Infinite Curvature: Understanding Black Holes

[pibomb]

What does it mean for something to have an infinite curvature (like a black hole?)?

 

[thepassenger48]

Black holes do not necessarily have to have infinite curvature, but enough mass to prevent light from escaping its gravitational pull. Infinite curvature would require an infinitely massive object; it's only a theoretical idea and is probably not possible. Einstein didn't believe that space time could rip, but only bend.

If it could exist, it would mean that the space-time continuum was bent to infinity. In case you don't know, the space time continuum can be thought of as a soft surface (such as a mattress), that curves when a mass is on it, and that gives us the impression of gravitational acceleration. It’s a pretty “sketchy” way of thinking about it, but it works.

 

[actionintegral]

I checked for the definition of curvature, thinking I could come up with a good answer. I found many definitions of curvature! So I am leaving this one alone

 

[pervect]

In GR, curvature referes to one of several "curvatuare tensors". The most fundamental of these is the Riemann curvature tensor - if you know the values of the components of the Riemann curvature tensor, you can calculate the values of the other tensors (most notably the Ricci, Einstein, and perhaps the Weyl).

A tensor is not just a single number, but an "array" of related numbers.

"Infinite" means that the value of at least one of the components of said tensor (remember, a tensor is not just a single number) increases without bound as one approaches a point, such as the singularity of a black hole. This is usually taken to imply that the theory itself breaks down and is not valid at that singular point.

unfortunately, there isn't any really super-simple way to describe what the Riemann curvature measures, though this is discussed in, for instance:

http://math.ucr.edu/home/baez/gr/outline2.html
http://math.ucr.edu/home/baez/einstein/node9.html